# Modeling Silicon-Dominant Anodes: Parametrization, Discussion, and Validation of a Newman-Type Model

^{*}

## Abstract

**:**

## 1. Introduction

_{1}5Si

_{4}or Li

_{2}2Si

_{5}, respectively. This is more than ten times the capacity of graphite [3]. At the same time, silicon has a comparable or even lower mass density than graphite [3], rendering it a potentially interesting anode material. However, the volume of silicon changes drastically during (de)lithiation, introducing several problems, e.g., mechanical cracking and disintegration, loss of electric contact, or breakdown of the solid-electrolyte interphase (SEI) [3].

^{−2}is application-oriented. Furthermore, the silicon particle diameter is in the micrometer range [11], which has hardly been studied in literature, especially not simulatively. Finally, it is important to mention that the silicon is partially lithiated to only approximately 30% (i.e., approximately 1200 mAh ${\mathrm{g}}_{\mathrm{Si}}^{-1}$ ). This benefits the mechanical integrity and thus the cycling stability of the electrode [11].

## 2. Experimental Procedure

_{0.8}Co

_{0.15}Al

_{0.05}O

_{1.985}(BO

_{3})

_{0.01}. As already mentioned, a more detailed description of further electrode properties related to the manufacturing step can be found in Section S1 along with Tables S1 and S2 of the Supplementary Materials.

^{®}(Solon, OH, USA) T-cells for validation. Coin cells were assembled as half cells vs. lithium (Li) metal as well as full cells. The data retrieved from the coin cells was used for characterization, especially to retrieve the half-cell equilibrium potentials using an electrode balancing process, which is discussed below. Additionally, Swagelok

^{®}T-cells using a three-electrode configuration were assembled as full cells with a Li metal reference electrode. The T-cells were used for validation at higher C-rates. Detailed information on the cell assembly process as well as the separator and electrolyte used are given in Section S2 of the Supplementary Materials.

## 3. Modeling

^{®}5.4. The solver settings are unchanged from the default solver settings recommended by the software for the given set of equations. The final parameters as used for the simulations are listed in Table A2. Some parameters are given by the cell design, such as the porosity or the active material volume fractions. Since these parameters are given, they are discussed in Section S1 of Supplementary Materials. The equilibrium potentials of the anode and cathode were measured in half-cell setups vs. Li metal. The remaining parameters are discussed based on a literature review, namely the electrolyte properties, solid-phase diffusivity, electrical conductivity, exchange current density, and charge transfer coefficients. A summary of these unknown parameters of the silicon electrode is given in Table 2.

#### 3.1. Equilibrium Potential

#### 3.2. Voltage Hysteresis

**Figure 1.**Quasi open-circuit potential of the measured Si and NCA half cell for the (de)lithiation as used in the model. The data are balanced to the full-cell potential (see Figure 2 and Figure A1) and transformed to refer to the calculated degree of lithiation (DoL) $\chi $ of the respective electrode. The degree of lithiation (DoL) is the capacity stored in the material relative to the fully lithiated state with ${C}_{\mathrm{gr},\mathrm{Si},\mathrm{th}}=3579$ mAh g

^{−1}and ${C}_{\mathrm{gr},\mathrm{NCA},\mathrm{th}}=279$ mAh g

^{−1}. Further information on the axis transformation process can be found in Section S5 along with Table S4 and Figure S1 of the Supplementary Materials.

#### 3.3. Electrolyte Properties

^{−1}. The electrolyte properties were assumed to be similar to th published by Valøen and Reimers [46] and only the ionic conductivity was scaled to match the supplier information.

#### 3.4. Electrical Conductivity

^{3}S m

^{−1}[47]. This is approximately four orders of magnitude greater than the values reported for the two active materials used in the present work. A composite electrode consisting of active material and graphite as a conductive agent would thus show an increase in its electrical conductivity when compared to pure active material. However, since the properties of such a composite electrode again depend on morphological parameters such as the porosity or the particle size distribution, and especially the volume or weight ratio between the two constituents, finding a definitive value for a given electrode can be challenging.

^{−1.5}S m

^{−1}to 1 S m

^{−1}, depending on the stoichiometric coefficient $\chi $ in Li

_{1-$\chi $}Ni

_{0.8}Co

_{0.15}Al

_{0.05}O

_{2}at 30 ${}^{\circ}\mathrm{C}$ [48]. For NCA composite electrodes, on the other hand, an effective value of 100 S m

^{−1}[49] has been reported. The presence of conductive agent in the porous electrode increases the electrical conductivity by approximately two orders of magnitude. Given the fact that the electrode used in the present work also contains conductive agent, the effective value of 100 S m

^{−1}is used here rather than the bulk value.

^{−4}S m

^{−1}[50] (p. 258), which can be increased by several orders of magnitude by doping [51] while following Arrhenius characteristics with increasing temperature [52]. Chockla et al. [33] for example manufactured silicon nanowires without any conductive carbon and measured an as-made electrical conductivity of 0.2 nS m

^{−1}. They could improve these values to 1400 nS m

^{−1}via annealing under a reducing atmosphere at elevated temperatures. McDowell and Cui [34] also investigated the electrical conductivity of nanowires without conductive carbon, but found values of around 400 S m

^{−1}. They found similar values of 600 S m

^{−1}for silicon thin films. An important difference is that Chockla et al. [33] could not even lithiate the untreated nanowire because of its low electrical conductivity, while McDowell and Cui [34] conducted their measurement in the lithiated state. The difference in conductivity depending on the DoL was reported by Pollak et al. [35], who found amorphous silicon thin films to be approximately 3.5 orders of magnitude more conductive in the fully lithiated state than in the delithiated state. The highest conductivity value they measured is approximately 5 × 10

^{3}S m

^{−1}. In another study, Kim et al. [36] found the electrical conductivity of bare $\u230010\mu \mathrm{m}$ Si particles to be 2 S m

^{−1}, which could be improved to 1.81 × 10

^{3}S m

^{−1}via copper deposition on etched Si particles. The conductivity of the untreated particles is two orders of magnitude smaller than for the nanowires and thin films investigated by McDowell and Cui [34]. This might indicate a dependence on the electrode morphology, but might also be due to different degrees of lithiation or different phase compositions of the silicon, e.g., crystalline or amorphous. Among the values already mentioned, Chandrasekaran and Fuller [23] assumed a bulk value of 33 S m

^{−1}for their Si electrode including conductive carbon. Wang [37] used this value in simulations and found it to be a suitable value via comparison to experiments. Hence, the electrical conductivity of 33 S m

^{−1}is also used in this work. Due to the high fraction of conductive agent, the true value of the electrode used in this work might be even higher.

#### 3.5. Solid-Phase Concentration

^{−1}and 279 mAh g

^{−1}, respectively, while the densities of 2336 kg m

^{−3}and 4730 kg m

^{−3}are supplier information. The density for Si is the density in the purely crystalline state, and the density for NCA was obtained from powder X-ray diffraction data. These density values yield concentration values of 311,943 mol m

^{−3}and 49,239 mol m

^{−3}, respectively. Note that these values are considered to be a good guess but will be adjusted in order to fit the model to the measurement data.

^{−3}instead of 311,943 mol m

^{−3}. However, since no volume change of the active material particles is considered in the present work, the change in the maximum solid-phase concentration in dependence on the DoL is of no further interest here.

#### 3.6. Solid-Phase Diffusivity

^{−14}m

^{2}s

^{−1}to 10

^{−17}m

^{2}s

^{−1}depending on the DoL. A value of 6 × 10

^{−15}m

^{2}s

^{−1}is used for the simulations in this work, i.e., the value at $3.77$ $\mathrm{V}$ vs. Li metal. This is because it is centered between the minimal and maximal cathode half-cell potentials for the cell setup used in this work ($3.0$ $\mathrm{V}$ and $4.4$ $\mathrm{V}$ NCA vs. Li metal).

^{−13}m

^{2}s

^{−1}to 10

^{−20}m

^{2}s

^{−1}[21,38]. A detailed and more recent literature review by Sivonxay et al. [39] suggests even higher values up to 10

^{−11}m

^{2}s

^{−1}. They investigated Li diffusion properties in Li

_{x}Si and Li

_{x}SiO

_{2}via first-principle calculations which yielded values between 10

^{−11}m

^{2}s

^{−1}and 10

^{−15}m

^{2}s

^{−1}. Wang et al. [21] compared numerical simulations with in situ lithiation experiments in order to determine the diffusion coefficient of Li in Si. They reported the coefficient in the amorphous as well as in the Li-rich phase after the formation cycle to be in the order of 2 × 10

^{−15}m

^{2}s

^{−1}. However, Bordes et al. [32] showed that fast Li diffusion pathways exist along defects. Such defects may occur due to stacking faults in a single crystal but can also occur along grain boundaries in polycrystalline Si particles. They propose that due to this phenomenon, Li diffusion may be much faster than in a single crystal without defects, regardless of the particle size. This suggests that, at least in polycrystalline particles as in this work, the diffusion coefficient might be greater than that reported by Wang et al. [21]. Nonetheless, their value of 2 × 10

^{−15}m

^{2}s

^{−1}is used for the simulations in the present work, as it has been validated against experimental data.

#### 3.7. Electrode Kinetics

^{−2}has been reported [40]. For NCA particles dipped in PVdF binder, the exchange current density during lithiation can reach higher values of up to 10 A m

^{−2}[55].

^{−3}A m

^{−2}to 10

^{5}A m

^{−2}can be found. Bucci et al. [38] reported a value in the order of 10

^{5}A m

^{−2}for silicon thin film electrodes of about 100 μm thickness. Lory et al. [19] assumed the exchange current density to be non-limiting and chose a constant value of 2 A m

^{−2}for their model. Note that, in their model, silicon is assumed to have no direct contact with the electrolyte because it is embedded inside carbon particles of 12 μm diameter. The value thus represents the exchange current density between carbon and silicon. Li et al. [41] derived values between 0.7 A m

^{−2}to 1.3 A m

^{−2}based on potentiostatic intermittent titration technique (PITT) measurements on silicon thin films of up to 1000 nm. Swamy and Chiang [20] found values of approximately 1 A m

^{−2}by fitting impedance data of 500 μm thick silicon wafers to an equivalent circuit model. Chandrasekaran et al. [22] report values of about 10

^{−2}A m

^{−2}to show a good fit between their single particle model with a diameter of 60 nm and experiments conducted using silicon nanowires with a pristine diameter of 89 nm [56]. Finally, Wang [37] fitted a multi-radius model to measurements on nanowires and found a value of 10

^{−3}A m

^{−2}to be a good fit for the exchange current density. This value is well below the reported values determined using measurements, which might indicate that some cross-dependencies due to model assumptions influence the value found via fitting. Thus, this value should be taken with care. None of the references stated a clear dependency on the DoL. With the exception of Chandrasekaran et al. [22], none of the references discussed charge transfer coefficients and assumed them to be symmetric, i.e., ${\alpha}_{\mathrm{a}}={\alpha}_{\mathrm{c}}=0.5$. Additionally, different sample geometries that have been investigated in the literature also make it difficult to get a true value for the exchange current density when only determined for a single state. This is due to the uncertainty in the active surface area of different samples, which depends on the sample morphology and undergoes changes due to volume expansion. Furthermore, the different measurement approaches used to determine the exchange current density might impede cross-reference comparisons. For simulations in this work, 1 A m

^{−2}is considered to be a reasonable initial guess because it is on neither end of the range found in the literature. For future simulation works, two different experimental results would be valuable: On the one hand, a consistent value for a single sample obtained via different measurement approaches would indicate under which conditions cross-reference comparisons are valid. On the other hand, an extensive investigation of various samples using a single measurement approach could serve as a baseline for a consistent data base.

^{−3}is assumed, i.e., the value in the equilibrium state. By assuming a constant value for ${c}_{\mathrm{l}}$, the impact of gradients during operation is excluded for the transformation from exchange current densities to reaction rates. Doing so, the transformed reaction rates are not dependent on ${c}_{\mathrm{l}}$ and thus the dependency of ${i}_{0}$ on ${c}_{\mathrm{l}}$ in Equation (A9) is not accounted for twice. Since ${c}_{\mathrm{s},\mathrm{surf}}$ may vary from the maximum value ${c}_{\mathrm{s},\mathrm{max}}$, the reaction rate constant for a constant exchange current density may vary, too. To avoid division by zero when ${c}_{\mathrm{s},\mathrm{surf}}$ becomes 0 or ${c}_{\mathrm{s},\mathrm{max}}$, ${c}_{\mathrm{s},\mathrm{surf}}$ is limited to the range $[0.01{c}_{\mathrm{s},\mathrm{max}};0.99{c}_{\mathrm{s},\mathrm{max}}]$ here. The reaction rate k is solved for using these values and then averaged so that it is no longer a function of ${c}_{\mathrm{s},\mathrm{surf}}$. Averaging is necessary so that the dependency of ${i}_{0}$ on ${c}_{\mathrm{s},\mathrm{surf}}$ in Equation (A9) is not accounted for twice. The averaged values are 2.95 × 10

^{−12}m s

^{−1}for silicon and 1.90 × 10

^{−11}m s

^{−1}for NCA. However, because of the uncertainty of the exchange current densities found in the literature, these values will be subject to a fitting process. The final values after the fitting can be found in Table A2.

## 4. Results and Discussion

#### 4.1. Open-Circuit Potential Measurements

^{−2}.

#### 4.2. Electrode Balancing

#### 4.3. Transformation to the Degree of Lithiation

^{−1}[5] is used. For the current gravimetric capacity, the current capacity from the battery cycler is divided by the active material mass of the sample. Additionally, unlike the concentration, the measured capacity is no absolute value but the integration of the electric current. An integration constant or rather an initial value is needed to retrieve a proper value for the DoL. This initial value is assumed to be the irreversible loss during formation and will be discussed in the following.

#### 4.4. Fitting of Solid-Phase Concentrations

#### 4.5. Rate Capability

_{ATLAB}

^{®}2021b aiming to minimize the difference between simulation and measurement data for a 1 $\mathrm{C}$ constant current charge and discharge. All other C-rates were not part of the fitting procedure. They serve as validation data and can be seen in Figure 5 and Figure 6.

^{−12}m s

^{−1}for the anode and 1.64 × 10

^{−10}m s

^{−1}for the cathode. The values are also summarized in Table A2. Before further inspecting the full-cell potential, the optimized reaction rates are checked for plausibility regarding the range from the literature. To this end, the optimized reaction rates are re-transformed to exchange current densities. This re-transformation is analogous to the transformation of the initial exchange current densities to reaction rates, as described above. The results show a re-transformed and averaged exchange current density of 2.228 A m

^{−2}for the Si anode and 9.108 A m

^{−2}for the NCA cathode. These values deviate by a factor of about 2 to 9 from the initial value of 1 A m

^{−2}for both anode and cathode, as taken from the literature. Considering a range of several orders of magnitude for the anode exchange current density, the fitting is considered to be within an acceptable range. For the NCA material, a value of approximately 9 A m

^{−2}lies at the upper end of the range given in the literature.

## 5. Conclusions and Outlook

- Si is partially lithiated to a similar degree as in this study and as such does not undergo further amorphization or crystallization.
- Model equations have not been adjusted, for example with respect to complex electrode kinetics, solid state diffusion, or volumetric changes.
- Voltage hysteresis is accounted for via two equilibrium curves rather than a dedicated hysteresis model.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Acronyms | |

AM | active material |

CC | constant current |

CV | constant voltage |

DoL | degree of lithiation |

DVA | differential voltage analysis |

GITT | galvanostatic intermittent titration technique |

Li | lithium |

LIB | lithium-ion battery |

NCA | nickel-cobalt-aluminum-oxide |

PITT | potentiostatic intermittent titration technique |

pOCP | pulsed open-circuit potential |

qOCP | quasi open-circuit potential |

RMSE | root-mean-squared error |

SEI | solid-electrolyte interphase |

Si | silicon |

SoC | state of charge |

Roman Symbols | |

C | capacity, Ah |

c | concentration, mol m${}^{-3}$ |

D | diffusivity, m${}^{2}$ s${}^{-1}$ |

$\frac{d\phantom{\rule{2.0pt}{0ex}}\mathrm{ln}f\pm}{d\phantom{\rule{2.0pt}{0ex}}\mathrm{ln}{c}_{\mathrm{l}}(x,t)}$ | activity, no unit |

${E}_{\mathrm{eq}}$ | equilibrium potential, V |

$\mathcal{F}$ | Faraday’s constant, 96,485 A s mol${}^{-1}$ |

i | current density, A m${}^{-2}$ |

${i}_{0}$ | exchange current density, A m${}^{-2}$ |

${j}_{\mathrm{n}}$ | pore-wall flux, mol m${}^{-2}$ s${}^{-1}$ |

k | reaction rate constant, m s${}^{-1}$ |

L | through-plane thickness, m |

${N}_{\mathrm{M}}$ | MacMullin number, no unit |

r | r-axis or r-dimension (pseudo dimension), m |

$\mathcal{R}$ | universal gas constant, 8.314 J mol${}^{-1}$ K ${}^{-1}$ |

${R}_{p}$ | particle radius, m |

T | temperature, K |

t | time, s |

${t}_{0}^{+}$ | transference number, no unit |

x | x-axis or x-dimension, m |

Greek Symbols | |

$\alpha $ | charge transfer coefficient, no unit |

$\chi $ | stoichiometry, no unit |

$\epsilon $ | volume fraction, no unit |

$\eta $ | overpotential, V |

$\kappa $ | conductivity, S m${}^{-1}$ |

$\phi $ | electrical potential, V |

$\rho $ | density, kg m${}^{-3}$ |

Subscripts & Superscripts | |

a | anodic |

app | applied |

c | cathodic |

eff | effective |

gr | gravimetric |

l | liquid phase |

max | maximum |

ref | reference |

s | solid phase |

surf | surface |

th | theoretical |

tot | total |

## Appendix A. Model Equations and Parameters

**Table A1.**Equations for the p2D model. Here, ${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}$ is the spatial gradient in the real dimension, i.e., along the through-plane direction of the cell stack, and ${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}r}$ is the spatial gradient in the pseudo dimesion, i.e., along the radius of the spherical active material particles.

Spatial gradients | ${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}=\frac{\partial}{\partial x},\phantom{\rule{1.em}{0ex}}\phantom{\rule{1.em}{0ex}}{\nabla}_{\phantom{\rule{-0.166667em}{0ex}}r}=\frac{\partial}{\partial r}$ (r in spherical pseudo dimension) | (A1) |

Mass balance | ${\epsilon}_{\mathrm{l}}\frac{\partial {c}_{\mathrm{l}}(x,t)}{\partial t}={\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}\left(\right)open="("\; close=")">{D}_{\mathrm{l}}^{\mathrm{eff}}\left(\right)open="("\; close=")">{c}_{\mathrm{l}}(x,t)$ | (A2) |

Mass balance | $\frac{\partial {c}_{\mathrm{s}}(x,t,r)}{\partial t}={\nabla}_{\phantom{\rule{-0.166667em}{0ex}}r}\left(\right)open="("\; close=")">{D}_{\mathrm{s}}{\nabla}_{\phantom{\rule{-0.166667em}{0ex}}r}{c}_{\mathrm{s}}(x,t,r)$ | (A3) |

Potentials | ${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{\phi}_{\mathrm{l}}(x,t)=-\frac{{i}_{\mathrm{l}}(x,t)}{{\kappa}_{\mathrm{l}}^{eff}}+\frac{2\mathcal{R}T}{\mathcal{F}}\left(\right)open="("\; close=")">1-{t}_{+}^{0}{\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}\mathrm{ln}{c}_{\mathrm{l}}(x,t)$ | (A4) |

${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{\phi}_{\mathrm{s}}(x,t)=-\frac{{i}_{\mathrm{s}}(x,t)}{{\kappa}_{\mathrm{s}}}\phantom{\rule{2.em}{0ex}}\mathrm{with}\phantom{\rule{2.em}{0ex}}{i}_{\mathrm{app}}\left(t\right)={i}_{\mathrm{s}}(x,t)+{i}_{\mathrm{l}}(x,t)\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}x,t$ | (A5) | |

Charge balance | ${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{i}_{\mathrm{l}}(x,t)+{\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{i}_{\mathrm{s}}(x,t)=0\phantom{\rule{2.em}{0ex}}\mathrm{with}\phantom{\rule{2.em}{0ex}}{\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{i}_{\mathrm{s}}(x,t)=-\frac{3{\epsilon}_{\mathrm{AM}}}{{R}_{p}}\mathcal{F}{j}_{n}(x,t)$ | (A6) |

Electrode kinetics | ${j}_{n}(x,t)=\frac{{i}_{0}(x,t)}{\mathcal{F}}\left(\right)open="("\; close=")">\mathrm{exp}\left(\right)open="("\; close=")">\frac{{\alpha}_{\mathrm{a}}\mathcal{F}\eta (x,t)}{\mathcal{R}T}$ | (A7) |

$\eta (x,t)={\phi}_{\mathrm{s}}(x,t)-{\phi}_{\mathrm{l}}(x,t)-{E}_{eq}(x,t)$ | (A8) | |

${i}_{0}(x,t)=\mathcal{F}{k}_{c}^{{\alpha}_{\mathrm{a}}}{k}_{\mathrm{a}}^{{\alpha}_{c}}{\left(\right)}^{{c}_{\mathrm{s},\mathrm{max}}}{\alpha}_{\mathrm{a}}{\left(\right)}^{\frac{{c}_{\mathrm{l}}}{\left({c}_{\mathrm{l},\mathrm{ref}}\right)}}{\alpha}_{\mathrm{a}}$ | (A9) | |

Effective transport parameters | $\left({\kappa}_{\mathrm{s}}^{\mathrm{eff}}\right)=\frac{{\kappa}_{\mathrm{s}}}{\left({N}_{\mathrm{M}}\right)},\phantom{\rule{1.em}{0ex}}{\kappa}_{\mathrm{l}}^{\mathrm{eff}}=\frac{{\kappa}_{\mathrm{l}}}{{N}_{\mathrm{M}}},\phantom{\rule{1.em}{0ex}}{D}_{\mathrm{l}}^{\mathrm{eff}}=\frac{\left({D}_{\mathrm{l}}\right)}{{N}_{\mathrm{M}}}$ | (A10) |

Boundary conditions | ${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{c}_{\mathrm{l}}(x,t){|}_{x=0\phantom{\rule{3.33333pt}{0ex}}\wedge \phantom{\rule{3.33333pt}{0ex}}x={L}_{\mathrm{tot}}}=0$ | (A11) |

${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{\phi}_{\mathrm{l}}(x,t){|}_{x=0\phantom{\rule{3.33333pt}{0ex}}\wedge \phantom{\rule{3.33333pt}{0ex}}x={L}_{\mathrm{tot}}}=0$ | (A12) | |

${\phi}_{\mathrm{s}}(x,t){|}_{x=0}=0$ | (A13) | |

${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}x}{\phi}_{\mathrm{s}}(x,t){|}_{x={L}_{\mathrm{tot}}}=-\frac{{i}_{\mathrm{app}}}{{\kappa}_{\mathrm{s}}^{\mathrm{eff}}}$ | (A14) | |

${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}r}{c}_{\mathrm{s}}(x,t,r){|}_{r=0}=0$ | (A15) | |

${\nabla}_{\phantom{\rule{-0.166667em}{0ex}}r}{c}_{\mathrm{s}}(x,t,r){|}_{r={R}_{p}}=-\frac{{j}_{n}}{{D}_{\mathrm{s}}}$ | (A16) |

Anode | Separator | Cathode | |
---|---|---|---|

Geometry | |||

Thickness coating L | 46 μm ^{m,c} | 2 × 260 μm ${}^{\mathrm{d}}$ | 68 $\mu $$\mathrm{m}$ ${}^{\mathrm{m},\mathrm{c}}$ |

${d}_{50}$ particle radius ${R}_{p}$ | 2.25 μm ^{L1} | 3.0755 μm ${}^{\mathrm{d}}$ | |

Active material volume fraction ${\epsilon}_{\mathrm{AM}}$ | 0.32 ${}^{\mathrm{c}}$ | 0.61 ${}^{\mathrm{c}}$ | |

Electrolyte volume fraction (porosity) ${\epsilon}_{\mathrm{l}}$ | 0.50 ^{c} | 0.55 ${}^{\mathrm{L}8}$ | 0.32 ${}^{\mathrm{c}}$ |

MacMullin number ${N}_{\mathrm{M}}$ | 5.385 ${}^{\mathrm{c}}$ | 1.29 ${}^{\mathrm{L}9,\mathrm{c}}$ | 7.333 ${}^{\mathrm{c}}$ |

Thermodynamics | |||

Equilibrium potential ${E}_{\mathrm{eq}}$ | Figure 1 ^{m} | Figure 1 ^{m} | |

Stoichiometry $\chi $ 100% SoC | 0.3068 ^{f} | 0.1396 ${}^{\mathrm{f}}$ | |

0% SoC | 0.0396 ${}^{\mathrm{f}}$ | 0.7954 ${}^{\mathrm{f}}$ | |

Max. concentration ${c}_{\mathrm{s},\mathrm{max}}$ | 322,067 mol m^{−3 c,f} | 46,400 mol m^{−3 c,f} | |

Transport | |||

Solid diffusivity ${D}_{\mathrm{s}}$ | 2 × 10^{−15} m^{2} s^{−1 L3} | 1 × 10^{−14} m^{2} s^{−1 L7} | |

Electric conductivity ${\kappa}_{\mathrm{s}}$ | 33 S m^{−1 L3} | 1 S m^{−1 L6} | |

Kinetics | |||

Reaction rate constant k | 5.78 × 10^{−12} m s^{−1 L2,f} | 1.64 × 10^{−10} m s^{−1 L4,f} | |

Anodic charge transfer coefficient ${\alpha}_{\mathrm{a}}$ | 0.5 ^{a} | 0.5 ^{a} | |

Cathodic charge transfer coefficient ${\alpha}_{\mathrm{c}}$ |
0.5 ^{a} |
0.5 ^{a} | |

Electrolyte * | |||

Salt diffusivity ${D}_{\mathrm{l}}$ in m^{2} s^{−1} *, ${}^{\mathrm{L}5}$ | ${10}^{-4}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-4.43-\frac{54}{T-(229+5{c}_{l})}-0.22{c}_{l}}$ | ||

Ionic conductivity ${\kappa}_{\mathrm{l}}$ in S m^{−1} *, ${}^{\mathrm{L}5,\mathrm{f}}$ | $0.5358\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-1}{c}_{\mathrm{l}}{(-10.5+0.668{c}_{\mathrm{l}}+0.494{c}_{\mathrm{l}}^{2}+0.0740T-0.0178{c}_{\mathrm{l}}T-8.86\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-4}{c}_{\mathrm{l}}^{2}T-6.96\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-5}{T}^{2}+2.8\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{-5}{c}_{\mathrm{l}}{T}^{2})}^{2}$ | ||

Activity $\frac{d\mathrm{ln}{f}_{\pm}}{d\mathrm{ln}{c}_{\mathrm{l}}(x,t)}$ (no unit) *,${}^{\mathrm{L}5}$ | $\frac{(0.601-0.24{c}_{\mathrm{l}}^{0.5}+0.982(1-0.0052(T-293.15)){c}_{\mathrm{l}}^{1.5})}{1-{t}_{+}^{0}}-1$ | ||

Transference number transference number${t}_{+}^{0}$ ${}^{\mathrm{L}5}$ | $0.38$ | ||

Reference concentration ${c}_{\mathrm{l},\mathrm{ref}}$ ${}^{\mathrm{a}}$ | 1 mol m^{−3} |

^{d, f}) or based on the literature (

^{L,f});

^{L}from the literature; ${}^{\mathrm{L}1}$ [13]; ${}^{\mathrm{L}2}$ [20]; ${}^{\mathrm{L}3}$ [37]; ${}^{\mathrm{L}4}$ [40]; ${}^{\mathrm{L}5}$ [46]; ${}^{\mathrm{L}6}$ [49]; ${}^{\mathrm{L}7}$ [53]; ${}^{\mathrm{L}8}$ [64]; ${}^{\mathrm{L}9}$ [65]; ${}^{\mathrm{m}}$ measured.

## Appendix B. Additional Measurement Results

**Figure A1.**Approximation of the quasi open-circuit potential (qOCP) under constant current discharge at $\mathrm{C}$/50 and 25 °C. Differential voltage analysis (

**c**) was used to reconstruct the measured qOCP vs. full-cell SoC (

**a**) under discharge of the full cell (see NCA+Si (FC)) via the (de)lithiation paths of the respective half cells (see NCA (HC) and Si (HC)). The approximation errors vs. SoC as well as the RMSE of the qOCP are shown in subplot (

**b**)).

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**Figure 2.**Approximation of the quasi open-circuit potential (qOCP) under constant current charge at $\mathrm{C}/50$ and 25${}^{\circ}\mathrm{C}$. Differential voltage analysis (

**c**) was used to reconstruct the measured qOCP vs. full-cell SoC (

**a**) under charge of the full cell (see NCA + Si (FC)) via the (de)lithiation paths of the respective half cells (see NCA (HC) and Si (HC)). The approximation errors vs. SoC as well as the RMSE of the qOCP are shown in subplot (

**b**)).

**Figure 3.**quasi open-circuit potential (qOCP) data (dashed lines) and pulsed open-circuit potential (pOCP) data (solid lines) for a current rate of $\mathrm{C}/50$ (black and gray) as well as $\mathrm{C}/10$ (red). The relaxed potentials of the 2 h open-circuit phases of the pOCP test were interpolated and are displayed as a gray/red area, indicating the voltage hysteresis effect between charging and discharging. The inset shows one exemplary open-circuit phase in more detail. The data are plotted as a function of the normalized charge capacity of the respective test. The tests were conducted at 25 ${}^{\circ}\mathrm{C}$.

**Figure 4.**Full-cell potentials during the two different fitting procedures compared to measurement data. “No fitting” refers to the initial parametrization taken from literature, “${c}_{\mathrm{s},\mathrm{max}}$ fitting” is an intermediate step, and “k fitting” refers to the final parametrization. Data are plotted as a function of the (dis)charge capacity normalized to the capacity during a $\mathrm{C}/50$ or $\mathrm{C}/10$ charge for both, the measurement (black) and model (colored), individually. Only CC phases are plotted, CV phases are not shown. (

**a**) shows the improvement due to the fitting of ${c}_{\mathrm{s},\mathrm{max}}$ at $\mathrm{C}/50$, and (

**b**) shows the improvement due to the fitting of k at $1\mathrm{C}$. The RMSE is that between simulation and measurement and is color-coded according to the legend.

**Figure 5.**Full cell potentials as validation data for the final parametrization after fitting using C-rates of $\mathrm{C}/10$ (

**a**), $\mathrm{C}/2$ (

**b**), and $2\mathrm{C}$ (

**c**). Color-coding and representation identical to Figure 4.

**Figure 6.**Rate capability of measurements using T-cells (black) and simulation results (red) for CC (dis)charge tests at C-rates of $\mathrm{C}/10$, $\mathrm{C}/5$, $\mathrm{C}/3$, $\mathrm{C}/2$, 1 $\mathrm{C}$, 2 $\mathrm{C}$, 3 $\mathrm{C}$, 5 $\mathrm{C}$, and 10 $\mathrm{C}$. The (dis)charge capacity of the CC steps is normalized to the capacity during a $\mathrm{C}/10$ CC charge for both, the measurements and the model, individually. The test was conducted between 2.8 V and 4.2 V at a constant temperature of 25 ${}^{\circ}\mathrm{C}$.

Ref. | Focal Area | Silicon | Dimensionality |
---|---|---|---|

[18] | Voltage hysteresis via asymmetric reactions | pure | 0D model |

[19] | Current distribution & inhomogeneous lithation | composite | p2D |

[20] | Kinetic limitations & rate capability | pure | 0D impedance model |

[21] | Phase boundaries & diffusivity | pure | 2D SPM |

[22] | Volume change & (de)insertion process | dominant | 1D SPM |

[23] | Volume change & electrolyte displacement | dominant | p2D |

[24] | Mechanical stress & stress-induced diffusion | pure | 1D SPM |

[25] | Mechanical stress & geometry dependency | dominant | 1D thin-film & SPM |

[26] | Coupling of electrochemistry and mechanics | pure | 2D MSM ${}^{\mathrm{a}}$ |

[27,28] | Capacity fade via SEI interaction | composite | 1D model |

[29] | Voltage hysteresis via mechanical stress | pure | 0D model |

[30] | First principles model of silicon lithiation | pure | 0D DFT calculation |

This work | Parametrization & validation of an electrode in a full-cell setup | dominant | p2D |

**Table 2.**Overview of silicon material parameters as can be found in the literature. The selected parameters are those model parameters that could not be retrieved otherwise.

Parameter | Unit | Value Range | Value Selected for This Work | References | ||
---|---|---|---|---|---|---|

Electrical conductivity | $\mathrm{S}{\mathrm{m}}^{\mathrm{-}1}$ | 0.2 × 10^{−9} | to | 5 × 10^{3} | 33 | [23,33,34,35,36,37] |

Solid-phase diffusivity | ${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ | 1 × 10^{−20} | to | 1 × 10^{−11 a} | 2 × 10^{−15} | [21,38,39] |

Exchange current density | $\mathrm{A}{\mathrm{m}}^{\mathrm{-}2}$ | 1 × 10^{−3} | to | 1 × 10^{5} | 2.2 ^{b} | [19,20,22,38,40,41] |

Charge transfer coefficient | – | 0.2 | to | 3.42 | 0.5 | [22,42,43] |

**Table 3.**Degree of lithiation and corresponding equilibrium potential at 0% and 100% SoC from Figure 1. The values are given for the full-cell charge (cha) and discharge (dis) as well as the averaged (ave) value at the respective SoC.

SoC | Anode | Cathode | |||
---|---|---|---|---|---|

DoL | ${\mathit{E}}_{\mathit{eq}}$ | DoL | ${\mathit{E}}_{\mathit{eq}}$ | ||

in – | in V | in – | in V | ||

0% | ${\chi}_{0,\mathrm{Si}}=0.0396$ | 0.7531 | ${\chi}_{0,\mathrm{N}\mathrm{C}\mathrm{A}}=0.7954$ | 3.5518 | dis |

0.7267 | 3.5635 | cha | |||

0.7399 | 3.5577 | ave | |||

100% | ${\chi}_{1,\mathrm{Si}}=0.3068$ | 0.1932 | ${\chi}_{1,\mathrm{N}\mathrm{C}\mathrm{A}}=0.1396$ | 4.3797 | dis |

0.1710 | 4.3764 | cha | |||

0.1821 | 4.3781 | ave |

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**MDPI and ACS Style**

Durdel, A.; Friedrich, S.; Hüsken, L.; Jossen, A.
Modeling Silicon-Dominant Anodes: Parametrization, Discussion, and Validation of a Newman-Type Model. *Batteries* **2023**, *9*, 558.
https://doi.org/10.3390/batteries9110558

**AMA Style**

Durdel A, Friedrich S, Hüsken L, Jossen A.
Modeling Silicon-Dominant Anodes: Parametrization, Discussion, and Validation of a Newman-Type Model. *Batteries*. 2023; 9(11):558.
https://doi.org/10.3390/batteries9110558

**Chicago/Turabian Style**

Durdel, Axel, Sven Friedrich, Lukas Hüsken, and Andreas Jossen.
2023. "Modeling Silicon-Dominant Anodes: Parametrization, Discussion, and Validation of a Newman-Type Model" *Batteries* 9, no. 11: 558.
https://doi.org/10.3390/batteries9110558